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MSc Physics and Astronomy

Advanced Matter and Energy Physics

Visualising the deformation behaviour of composite materials

by

Melín Dominique Erin Walet

10564551 Master Thesis

60 ECTS

November 2018 - May 2020 Supervisor:

prof. dr. P. (Peter) Schall

Daily supervisor:

J. (Joep) Rouwhorst MSc.

Second examiner:

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The deformation mechanism of soft condensed matter have been a highly relevant research topic over the past few years. Amongst others, this is of great importance for the food indus-try, where products are required to have specific properties. Familiar products such as bouillon cubes are known to exhibit complex behavior as a response to external stress. Due to the combination of viscoelastic properties and the presence of hard inclusions, the response of this kind of composite materials is far from trivial. Previous research has proven the existence of universal deformation behavior across many length scales. Scaling laws in force drop size dis-tributions permit comparison of experimental laboratory results with larger scale events such as earthquakes. Moreover, clustering of local strain areas was observed in colloidal glasses. However, further investigation is needed to comprehend the deformation mechanism in soft composite materials. In this thesis we investigate the deformation mechanism of a model soft composite: cube-shaped viscoelastic matrices containing a controlled concentration of hard inclusions. We perform indentation experiments and measure the exerted pressure while at the same time recording images of the fluorescently dyed matrix around the inclusions. We test volume fractions ranging from 0 to 40% and find power-law force drop size distributions. This implies a universal deformation behavior, independent of the presence and volume frac-tion of hard inclusions. Furthermore, with the aid of the direct images, a displacement and strain field is obtained in planes both parallel and perpendicular to the indentation direction. Clustering of local strain areas occurs on a small scale. We compare our experimental findings with theoretical predictions for purely elastic materials. A Hertzian-like distribution is found. However, more strain peaks occur deeper inside the sample which correlate with the presence of local strain areas.

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Contents

1 Introduction 1

1.1 Soft composites . . . 1

1.2 A model for soft composite materials . . . 3

2 Theoretical Background 5 2.1 Introduction to material deformation . . . 5

2.1.1 Stress and strain . . . 5

2.1.2 Stress tensor and principal stresses . . . 5

2.1.3 Nonaffine displacements . . . 7

2.1.4 Hertzian elastic strain distribution . . . 8

2.2 Viscoelastic deformation mechanics . . . 10

2.2.1 Elastic versus viscous behavior . . . 10

2.2.2 Universal scaling laws . . . 11

2.2.3 Force drops and avalanche statistics . . . 11

2.3 Viscoelastic composite deformation mechanics . . . 12

2.3.1 The effect of inclusions in a material . . . 12

2.3.2 Local strain clusters . . . 13

3 Materials and methods 15 3.1 Materials . . . 15

3.2 Experimental Methods . . . 15

3.2.1 Gel cube production . . . 15

3.2.2 Setup . . . 17

3.2.3 Experimental settings . . . 18

3.3 Data retrieval and analysis methods . . . 19

3.3.1 Data Retrieval: Pressure readings . . . 19

3.3.2 Data Retrieval: Real-space data . . . 19

3.3.3 Pressure readings and force drop size distributions . . . 19

3.3.4 Preprocessing Real-Space Data . . . 20

3.3.5 Particle locating and tracking . . . 20

3.3.6 Local strain field . . . 22

3.4 Discussion of methods . . . 23

4 Results 25 4.1 Force drop size distributions . . . 25

4.2 Real-space deformation of the beadless cube . . . 27

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4.2.2 Taking a perpendicular view . . . 29

4.3 Filling the cube with 0.1 VF inclusions . . . 30

4.3.1 2D strain distributions . . . 30

4.3.2 Fluctuating strain field in 3D . . . 31

4.4 Increasing the PMMA inclusions to 20% . . . 34

4.4.1 2D strain distributions . . . 34

4.4.2 More strain fluctuations in the perpendicular plane . . . 35

4.5 Clustering of local strain regions . . . 38

4.6 Comparison to Hertzian theory . . . 39

5 Conclusion 44

6 Outlook 44

Acknowledgements 45

Populair wetenschappelijke samenvatting 47

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1

Introduction

How does sand respond under the weight of your feet during a walk on the beach? What happens to the toothpaste inside a tube when being squeezed? With the most basic actions, you can wonder about the physics behind it. Did it ever cross your mind how easily you can spread the margarine on your toast for breakfast? Or how bouillon cubes crumble between your fingers when making broth? Questions such as the above all come down to the field of soft condensed matter physics.

Materials with specific deformation behavior are vital for industrial and medical applications. Bouillon cubes should crumble effectively between your fingers and the structure of your low-fat margarine should make it effortless to spread. However optimizing the mechanical behavior of food products is difficult: products have to live up to certain health standards while be-ing uncompromisbe-ing in taste and structure. To quote a recent publication of Unilever R&D:

’Unilever is committed to ensuring that by 2020, 75% of our foods products adhere to the daily salt limit of 5g, recommended by the World Health Organization.’1

1.1

Soft composites

A thorough understanding of the deformation behavior of purely elastic materials has been developed previously. However, viscoelastic materials which show solid-like as well as fluid-like properties are not yet fully understood. Furthermore, soft composite materials that consist of a viscoelastic matrix with hard or soft inclusions, show even more complex deformation. In the past decades many researchers have focused on describing the interesting behavior of these materials.2,3 In this thesis we deal with composite materials which are viscoelastic matter

with soft or hard inclusions. The inclusions introduce a second, larger length scale to the material, that interferes with smaller scale strain correlations in the viscoelastic matrix. For some viscoelastic materials, the heterogeneous structure can only be seen on a certain length scale which is small enough (micro-scale). Yet, the hard inclusions introduce another length scale which interferes with the deformation of the viscoelastic matrix.

Well-known examples of composite materials are filler-reinforced rubber used in car tyres, constructions of composite wood and composite implants for medical applications.4 These

composite materials are manufactured to create specific performance characteristics such as reduced roadway noise, improved toughness or increased breaking efficiency of your car tyres. Soft composites are encountered on various length scales, varying from hybrid carbon nan-otube composites,5 to filler reinforced car tyres and even bouillon cubes (Figure 1).

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conduc-(a) (b)

(c) (d)

Figure 1: Composite materials are manufactured for various industrial applications, such as micro-silica fillers5 in carbon nanotube composites with two different types of fillers (1a), filler-reinforced car tyres (1b),6 bouillon cubes (1c), and clay agglomerates (1d).7

tivity.5 By using particulate filers, the strength and toughness can be increased effectively.

Another example of composites are clay agglomerates in geomaterials8 (Figure 1d). These

micro-composites are used for building and consist of large sand grains embedded in a clay matrix phase.7 The macroscopic strength due to the binding properties of the clay matrix can

be increased with natural tannins. An important effect is the aging of the material. Induced by failing of the internal binding, this aging effect will weaken the macroscopic strength. Composites can also be found in medical applications. Prostheses can be created with com-posites to mimic biological tissues. For example, a composite implant of a coral and collagen had been tested for the use in the healing of bone defects.4

Moreover, we encounter many soft composites in the food industry. Bouillon cubes consist of a viscoelastic fat matrix and hard inclusions such as minerals. In order to change or optimize certain characteristics of the product, it is necessary to fully comprehend the deformation mechanism.

Previous research has shown that viscoelastic materials under increasing stress, can show local plastic events. This is observable by sudden drops in the force signal. These sudden force drops follow the same power-law scaling over a range of different systems.9 In 2009, a model

was introduced by Dahmen et al.10 which predicts this observed universality in slip statistics.

This universal behavior allows us to experiment at a small scale (in laboratory settings) and to link the results to events on a larger scale. This way we increase our understanding of large scale deformation such as earthquakes. Uhl et al. elaborated on this universal model in the work published in 2015.9 The model shows similarities with the Gutenberg-Richter law in

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brittle earthquakes statistics.11

A particularly interesting viscoelastic response to deformation occurs in colloidal glasses, where local strain region develop which evolve into larger, connected areas.12 The question arises

whether and how local strain areas cluster on a larger scale such as in composite viscoelastic materials with hard inclusions.

1.2

A model for soft composite materials

In this thesis, we study the deformation mechanism of viscoelastic composite materials using a model system. The model material consists of a cube-shaped viscoelastic vegetable fat matrix with a dispersion of hard inclusions (PMMA beads). A picture of the model system can be seen in Figure 2. We then use a hemispherical tip, which is a commonly used deformation geometry, to push into the model material. We investigate the evolution of local strain regions inside the cube. By comparing both the force drop size distribution (with data obtained by pressure sensors) as well as the strain evolution (with the aid of real-space images of the cube interior), we obtain insights in the role of hard inclusions on the deformation events. Specifi-cally, we look at the occurrence of force drops and analyze their size distribution. We explore the displacements of individual features inside the material to understand which local events take place.

(a)

10 mm

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Figure 2: Model composite system: a cube-shaped viscoelastic vegetable fat matrix with a dispersion of hard inclusions (PMMA beads). An image of the cube can be seen in Figure 2a. A 2D image from the interior of the cube during indentation is depicted in Figure2b.

This thesis is outlined as follows: theoretical background is given in Section 2, by in-troducing relevant terminology as well as elaborating on the results of previous research to familiarize the reader with the subject of soft matter indentation. In Section 3, details on the used materials and the experimental setup are given. Furthermore, a passage is devoted to the relevant analyzing principles, such as force drop size distributions and strain field calculations.

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In Section4, the results of the conducted indentation experiments are presented and discussed. Both the force drop size distributions, as well as a local maximum and compressive strain field in 2D sections parallel and perpendicular to the indentation direction are presented. Also, the existence and clustering of local strain areas inside the material is analyzed. Moreover, a comparison to the predicted Hertzian elastic strain distribution is made.

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2

Theoretical Background

In this Section, relevant theoretical background is provided about the indentation of composite viscoelastic matter. Basic principles on deformation mechanics are introduced in Subsection2.1, which is specified to the area of viscoelastic deformation mechanics in Subsection 2.2. Lastly, we focus on the deformation of composite viscoelastic matter in Subsection 2.3.

2.1

Introduction to material deformation

2.1.1

Stress and strain

When a material is subject to an external force, two kinds of changes will occur. On the one hand, rigid body translation or rotation will cause the object to move in its completeness. On the other hand, internal deformations can take place inside the object. A material can be deformed in several ways, such as by shearing, compressing or stretching. In all cases, the material is subject to an external force. Whether and how the material reacts to this force, depends greatly on the material properties. In this thesis, we will use two ways of investigating the deformation mechanism. Firstly, pressure sensors can be used to measure the force as a function of applied strain, see subsection 2.1.1 for corresponding theoretical definitions.

A relevant aspect to consider during indentation is the correlation between stress and strain. Stress is the force (in Newton) per unit area (m2), and can also be seen as the pressure

(in Pascal), exerted on the material. Strain is the deformation of the material relative to the original dimensions: (L−L0)

L0 , where L is the length after the deformation and L0 is the original

length. Therefore, strain is unitless.

For a 3-dimensional system, the stress tensor consists of nine components. Important to note is that the stress tensor is symmetric (σxy = σyx) which will simplify stress-strain equations.13

     σxx σxy σxz σyx σyy σyz σzx σzy σzz     

2.1.2

Stress tensor and principal stresses

Different stresses can be exerted on a body, such as the shear stresses (green lines in Figure

3) and the tensile stresses (blue lines). However, we can change the coordinate system such that the directions of the stress are normal to the new coordinate axis (x’ and y’ in Figure 3). The corresponding stresses, called principle stresses, are the eigenvalues of the diagonalized

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Figure 3: The principal stresses in a body. The corresponding coordinate system is indicated by x’ and y’ (right image).

stress tensor. The 2D stress tensor σ can be calculated in 2D in the following way:

σ0 = QσQT (1)

where Q is the transformation matrix:

  cos θP sin θP − sin θP cos θP  

with θP the principle orientation, which is calculated by solving the transformation equation

for τxy0(as described thoroughly by Gerre and Goodno, 200814). We set τxy0 to zero (since

the shear component should be zero in the principle planes) to obtain a value for the principle stress angle θP:

τxy0 = (σyy− σxx)sin(θP)cos(θP) + τxy(cos2(θP) − sin2(θP)) = 0 (2)

which leads to the following expression for θP:

tan(2θP) = 2τxy σxx− σyy (3) θP = 2Txy σxxσyy (4)

Eventually, the maximum shear stress can be calculated from the maximum and minimum principle strain values:

τmax =

|σmax− σmin|

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In a 3D system three combinations of the stress components can be ascribed, which do not depend on the coordinate system. Therefore, they are called stress invariants:

I1 = T r(σ) = σkk (6) I2 = σxx σxy σyx σyy + σxx σxz σzx σzz + σyy σyz σzy σzz

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I3 = Det(σ) = ijkσixσjyσkz (7)

Their physical interpretation is as follows: I1 is the hydrostatic component, I2 is related

to the deviatoric components (or Von Mises stress) and I3 provides information on the ratio

between deformed and initial volume of the material.

The value for the elastic moduli can differ in anisotropic materials, since then the material properties depend on the direction. This is why one needs the fully generalized Hooke’s law12

to describe anisotropic materials, where the elastic modulus is not a number but a tensor. The Generalized Hooke’s law can be written as:12

σij = Cijklkl (8)

where Cijkl are the components of the 4th order stiffness, or elastic moduli. For a isotropic

linear elastic material, we use the following expression for the elastic moduli:

Cijkl = λδij + µ(δikδjl+ δilδjk) (9)

to get the following relation between stresses and strains:

σij = λδijkk+ µ(ij + ji) (10)

2.1.3

Nonaffine displacements

Not only by investigating the measured force as a result of applied strain, but also by following the internal, local displacements of a material, insights in the deformation mechanism can be obtained. As a reaction on external stress, the strain can be equal throughout the material: this is called homogeneous deformation. The strain can also be inhomogeneous (nonaffine). In Figure 4 a schematic example is given of nonaffine displacement: the green circles follow the same direction and therefore deform affinely. The red circle however, shows nonaffine deformation.

Nonaffine deformations have also been studied for applications in biological physics, for instance by Hepworth et al. in 2001 and Wen et al. in 2012.15,16

It is possible to deduce the strain from deviations in local displacements17 by constructing

a strain tensor from the displacements in subsequent frames. More details are provided in Section 3.

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Figure 4: Schematic example of nonaffine displacement. The green circles deform in an affine way as a response to the applied force F. However, the displacement of the red circle diverges from its neighbours and is therefore called nonaffine.

2.1.4

Hertzian elastic strain distribution

Whenever two objects are in touch with each other, the contact radius, or Hertzian contact radius becomes an important characteristic to describe the stress inside a material. Considering two curved objects being pushed together, one could expect the pressure to rise to infinity. However, a small contact area is created between the two objects.18 This limits the stress

significantly. This radius depends on parameters such as the diameter and elastic modulus of both objects. The Hertzian theory predicts the location of the maximum local strain inside the material. Therefore, we will use this theoretical prediction to compare with the observed strain distribution from indentation experiments described in this thesis. Many scientists have worked with predictions derived from the Hertzian theory, and the theory has been compared to results from for example finite element analysis by Purushotohaman in 2014.19 Usage of the Hertzian

theory is only justifiable, when the following assumptions on the material configuration can be made.20

• The surfaces should be continuous and nonconforming,

• The indenter and the indented object can be seen as an elastic half-space, such that the contact radius is much smaller than the characteristic radius of the body,

• The bodies should be in friction-less contact,

• and lastly, the strains associated with the deformation should be small.

With the above-mentioned assumptions in mind, we consider the case in which a sphere indents an elastic half space. The radius of contact area can be described by the following equation.20 a ∼= 3RF 2E∗ 13 (11)

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with a the radius of contact area, R the radius of the indenter, F the externally applied force and E* satisfying the following equation:

1 E∗ = 1 2 1 − v2 1 E1 + 1 − v 2 2 E2  (12) with E1 and E2 the elastic moduli of the materials and v1 and v2 the corresponding Poisson

ratios.

Combining the expressions above, we are left with the following expression for the contact radius: a ∼= 3RF 4 1 − v2 1 E1 + 1 − v 2 2 E2 13 (13) For a purely elastic material being indented by a sphere, the maximum strain will be lo-cated at around 1/2 of the contact radius below the indentation tip according to the Hertzian theory, for a specific Poisson ratio of 0.33.12

Let us review the assumptions of Hertzian theory and inspect their validity for our exper-imental setup: a gel cube indented by a hemispherical indenter. Indeed, the surfaces of the touching bodies are continuous and nonconforming. Also, the gel cube and the indenter both have a radius undoubtedly larger than the contact radius. Moreover, the contact between the bodies is considered to be frictionless.

The strain should be small such that the deformation is still in the elastic regime. We will therefore only regard the first part of our indentation, during which the strain is low enough and the deformation is still elastically dominated.

Considering our indenter is made of a hard material (steel), the elastic modulus E2 rises to

infinity which allows us to neglect the 1−v22

4E2 term. This leaves us with the following expression

for contact radius a.

a ∼= 3RF 4 (1 − v2 1) E1 13 (14) Similar to the technique used by Rahmani in Chapter 6 of Micromechanics and rheology

of hard and soft-sphere colloidal glasses,Rahmani2013), we are able to visually determine the

size of the contact radius. By analyzing the real-space data at a certain point during the indentation, we are able to retrieve a value for the contact radius. Using this value we can estimate the location of the maximum strain.

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2.2

Viscoelastic deformation mechanics

2.2.1

Elastic versus viscous behavior

As mentioned above, the deformation behavior of viscoelastic material is rather complex due to the combination of elastic and viscous components.

Purely elastic materials deform reversibly under pressure. This implies that the material returns to its original shape, when the external force is released. Energetically speaking, no internal energy of the system is dissipated during the deformation.21 In any elastic regime, Hooke’s

law determines the relation between stress (σ) and strain ():

σ = E (15)

where E is the (constant) elastic modulus, σ is the stress and  is the strain. The elastic modulus is hence the ratio between the strain () and the applied stress (σ). Figure 5a shows the elastic stress-strain relation.

In the viscous part of the deformation, the stress depends on the strain rate and energy is dissipated during the loading and unloading of the system (see Figure5b). Simple viscoelastic models are the Kelvin Voigt model and the Maxwell model22where the strain rate is controlled

by the viscosity of the material.

(a) Purely elastic stress (σ)

versus strain () response.

(b) Viscous stress-strain

re-sponse. The stress depends on the strain rate and energy is dis-sipated during the process.

(c) Viscoelastic stress-strain

curve with local force drops (in black) occurring in the elas-tically dominated regime (red line).

Figure 5

.

Permanent deformation can occur when the stress exceeds a certain yield stress due to internal rearrangements inside the material. One can take a rubber band as an example: when stretching the band carefully, the strain increases proportionally to the applied stress and no permanent damage occurs. Overstretching however, will cause the band to break and therefore show permanent deformation.

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In Figure 5c these plastic events are visualized by the black lines.

2.2.2

Universal scaling laws

Over the past years, the existence of universal power-laws in the plastic deformation of mate-rials was investigated throughout various situations and length scales.23–26 Such universality is

of great importance, since it enables linking results from the lab down to nanoscale and up to large scales such as in earthquakes. Linking the macroscopic applied force to the microscopic internal fluctuations has long been considered to be a highly challenging task in material sci-ence.27 To overcome this challenge, granular and soft matter was investigated more easily

due to their properties of having easily measurable motion of individual particles. Moreover, it became possible to measure the applied force and fluctuations with excellent temporal res-olution. In order to explain the deformation mechanisms, Uhl et al.9 studied the correlations

of stress fluctuations.

In 2005, Schorlemmer showed that even earthquakes revealed strongly correlated deformations, by looking at the variations in earthquake size distributions across different stress regimes.28

In 2009, Dahmen and her colleagues presented a universal micro-mechanical model for the deformation in solids.10 This model offered a useful framework, which connects particle-scale

dynamics to macroscopic stress-strain response. In 2015, Dahmen et al. revealed the existence of similar scaling behaviour of the slip size distributions across various length scales.9 This

would imply universal deformation behavior, irrespective of size, from earthquakes to deforma-tions on nanoscale. First experimental proof on critical internal scaling reladeforma-tions was given by Denisov et al. in 2016,27 by linking the macroscopic applied force to the microscopic internal

fluctuations. The origin of force fluctuations appeared to lay in the internal strain distribution, with a diverging correlation length. The size distribution depended on both the shear force magnitude and the confining pressure. Importantly, this dependence can be accounted for by a scaling model. Within this model, detail-independent universal scaling exponents are present.9

The experimental proof agreed with the predictions of the mean field theory. These studies opened the door to a lot of yet unsolved questions about the deformation mechanisms. Denisov et al.29 showed for the first time that not only the scaling exponents but also the

scaling functions for the slip dynamics may be general across scales and structures. In 2017 Denisov30 eventually proved that both the statistics and dynamics were independent of scale

in bulk metallic glasses and granular matter.

2.2.3

Force drops and avalanche statistics

To investigate whether we encounter universal power-law distributions as described in Section

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Figure 6: Schematic overview of the measured force as a function of time. Force fluctuations are annotated in red and cause the serrated character of the curve. The inset shows the force drop size distribution on a log-log scale. The distribution follows a -3/2 power-law cutoff.

force (stress) versus time signal, we can observe local fluctuations, which imply plastic events. Such rearrangements inside the material causes the stress to decrease temporarily. This abrupt decrease is followed by an elastic force increase, resulting in a serrated force versus time signal. Figure 6shows a schematic example of the force fluctuations as a function of time.

By plotting the distribution of the force drop sizes, the behavior of the deformation can be investigated, see the inset of Figure 6. The sharp drops in the force can be accounted for by internal relaxation of the material, since this would release some of the applied force. With the size of the force fluctuations called s, the relative frequency P(s) can be plotted (see the inset of Figure 6). In previous research, power-law distributions were found with an exponent of -3/2.27

2.3

Viscoelastic composite deformation mechanics

In this research we examine a viscoelastic material with PMMA inclusions. The fact that these beads have completely different properties (such as its elastic modulus) than the viscoelastic gel matrix, makes the deformation even more complex. This inhomogeneity of the elasticity inside the material will have a significant effect on the deformation.31

2.3.1

The effect of inclusions in a material

In the past, scientists started to investigate the influence of inclusions on the deformation of a material. Already in 1957, a theoretical model was introduced to describe the elastic behavior of filler-reinforced vulcanized rubbers.32 The reinforcement of these rubber composites is

re-lated to the formation of a filler network.33 Moreover, Eshelby addressed the question how the

field of an elastic solid is affected by the presence of an elastic ellipsoidal inclusion in 1959.34

Later, other scientists proposed various models around this Eshelby inclusion problem.35–37 In

1973 the Mori-Tanaka micromechanical scheme was proposed to provide a relation between the average strains in the inclusions and the matrix phases.38 These models treat the deformation

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behavior of viscoelastic matrices with elastic or plastic inclusions. This Mori-Tanaka model has been applied to several specific geometries such as a transversely isotropic fiber reinforced UD composite.39 In 2010, Rendek investigated the influence of filler particles in filler-reinforced

rubber and found that the stress signals were strongly influenced by the amplitude-history of the deformation input.35 This resulted in the development of a new model for finite

viscoelas-ticity.

An additive interaction law for the Eshelby inclusion problem was derived.40 Subsequently,

Meguid et al. introduced a time-incremental Eshelby-based homogenization scheme for vis-coelastic heterogeneous materials in 2017.40 Constantinides et al. provided a framework to

determine information about the composite microstructure8 and to study the mechanical

per-formance of a given composite material. It was stated that the indentation depth should be small compared to the characteristic length scale of the sample in order to be able to access the elastic properties of the individual components. Therefore we will investigate the small-strain part of our indentation experiments as to examine the elastically dominated regime.

Homogenization schemes for aging linear viscoelastic materials with elongated inclusions were introduced by Laverge in 2016.41 It was shown that the viscoelastic behavior of concrete is not

significantly affected by the aspect ratio of aggregates. Huang presented a micromechanical approach to simulate the damage-couple viscoelastic response of an asphalt mixture.42 It was

shown that the damaged material has a larger stress distribution due to the irregularity of the coarse aggregates. The von Mises stress distribution becomes smaller as the loading time increases.

2.3.2

Local strain clusters

As pointed out by Weeks in his work Soft Jammed Materials,43 local weak spots inside a

material can have a huge effect on the deformation mechanism. Whether these weak spots refer to dislocations (in crystals) or shear transformation zones (in glasses and granular materials44),

the general consequence is the development of slip avalanches, when the slipping of a single weak spot triggers other weak spots inside the material. Shear transformation zones are described by various scientists17,45 in order to describe the region inside amorphous solids

where irreversible deformation occurs.

An interesting discussion point on shear transformation zones is addressed in the paper by Gendelman.45 The paper shows evidence that plastic events depend on the external loading.

This view is opposite to the more phenomenological belief that the events can be ascribed to certain regions inside the material.46,47

It was found by Chikkadi et al. that long-range strain correlations played a central role in the flow of glasses.48 New insights were offered on the shear-banding instability of not only

glasses, but of a wide range of amorphous materials.49 Experimentally, long-range strain

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studied by Rahmani. The system of soft spheres led to a higher yield strain and more gradual yielding, compared to the hard-sphere suspensions.3 Long-range strain correlations were found

throughout the indented material. In this thesis, we will investigate whether similar long-range strain correlations occur on a larger scale in composites such as bouillon cubes.

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3

Materials and methods

Indentation experiments were conducted on viscoelastic gel cubes containing a vary-ing amount of hard inclusions. In particular, the aim was to represent bouillon cubes (by viscoelastic gel cubes) with a varying amount of minerals (the hard inclusions). This chapter provides information about the materials, setup and analyses for the indentation experiments. In Section3.1 the materials are presented which were used for the production of gel cubes. In Subsection 3.2follows the step-by-step procedure of the gel cube production. Also, the setup of the indentation experiments is de-scribed. Lastly, the methods used to analyze the results are discussed in Subsection 3.3.

3.1

Materials

Fat-based oil gels were produced by mixing sunflower oil with gelling agents β-sitosterol (an edible plant sterol) and γ-oryzanol (an edible plant sterol ester). These sterols were provided by Unilever B.V.. The sunflower (Spar Zonnebloemolie) was purchased at a local supermarket. Moreover, Rhodamine B was added to the gel (from Sigma Aldrich) for imaging purposes. Spherical polymethylmethacrylate (PMMA) beads, with a diameter of four (4) mm, were added to the gel cubes to represent hard inclusions. These beads were produced by Spherotech - Precision Balls.

3.2

Experimental Methods

3.2.1

Gel cube production

Based on results of previous experimental trials, it was decided to create cubes with a volume percentage of beads (φinc) between 0.0 and 0.4.

The gel cubes were built up in horizontal layers of about 10 millimeters each. A cubic construction with dimensions of 50 by 50 by 50 mm was used as a mall for the cubes. This construction consisted of plates, which could be removed separately after full gelation of the cube. Figure 7a shows a picture of a cube inside its ’mall’ during gelation, and Figure 7b

shows a finished gel cube with 0.4 VF (volume fraction) of beads.

In the first part of the gel cube production process, the two gelling agents β-sitosterol and

γ-oryzanol were added to sunflower oil to get an mixture with 6 mass percent of sterol mixture

in oil. The mass ratio between β-sitosterol and γ-oryzanol was 2:3. The precise composition of the gel cubes was based on information provided by Bot et al.50,51

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(a) Picture of a cube inside its mall during the

gela-tion process.

(b) Picture of a finished cube with 0.4 VF of beads.

Figure 7: Pictures of the gel cubes used for the indentation experiments

.

The oil mixture was heated and stirred at 85 degrees C for at least 3 hours to dissolve the plant sterols. Lastly, fluorescent dye (Rhodamine B) was added. The heating and stirring thereafter continued for 20 minutes at the same temperature. This way, dye sprinkles are distributed in locally high concentrations throughout the cube (and therefore still visible for data analysis). Thereupon the mixture was cooled down and stirred again for about 30 minutes before being poured into a mall. By stirring the mixture before pouring it, the dye sprinkles are prevented from sedimentation. For imaging purposes it is of great importance that the dye is distributed homogeneously throughout the oil mixture whilst still being distinguishable as dye sprinkles.

The gelling process is a nucleation based process.51 During this process, tubules aggregate

and form an organogel. Edible sterols were used as gelling agents. β-sitosterol is an edible plant sterol whose structure resembles the one of cholesterol. It is used mostly in the food industry. γ-oryzanol is an edible plant sterol ester. As described by Bot and his colleagues,51

a mixture of these two sterols shows very different behavior than its individual components. The plant sterols form hollow tubes with a diameter of 7.2 ± 0.1 nanometer. According to Bot et al., the most likely growth mechanism for the nanotube is via a helical ribbon.50 When

heated up to a temperature from 85 degrees Celsius, the sterols dissolve in the oil. Therefore, the mixture will be in a liquid phase at such high temperatures. On the other hand, decreasing the temperature starts the gelling process.51 More information on the role of the sterols on

the gelling process in the papers by Bot et al.50,51

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was added. By throwing the beads from different heights and with different force, the beads will be located at different heights within the layer. The degree of gelation plays a role in this delicate process: the beads have to be able to find their way downwards through the oil mixture without sinking to the bottom.

Each layer was gelled in about 24 to 48 hours. For each additional layer, a new oil mixture was produced (to ensure the dye is not fully resolved, as would be the case in an ’old’ mixture). The temperature difference between the fresh mixture and the gelled layer is controlled, such that the upper part of the underlying layer partially melts. This way, the beads in the area get somewhat better distributed, while preventing the layer from melting completely. Due to this melting and regelling process, the tubules might be less strongly connected in the area at the interface. Therefore it should be noted that this area might be more fragile than areas in the middle of a layer.

3.2.2

Setup

In order to conduct the indentation experiments on gel cubes, the following setup was used. A gel cube was placed in a glass reservoir (with dimensions of approximately 50 cm by 30 cm by 30 cm) filled with sunflower oil. Two lasers were placed on both sides of the container, facing each other. This way, the entire cube could be scanned. Perpendicular to the lasers, a camera was placed. Thus, all information of the 3D cube was collected. A hemispherical indenter was located above the reservoir. A program called Shear Cell controlled the speed and direction of the indenter movement. Figure8shows a schematic representation of the setup and a picture of the actual setup of the indentation experiments.

(a) Schematic setup. (b) Picture of the actual setup.

Figure 8: The setup consists of a aquarium filled with sunflower oil, in which the cube is placed. On both sides, lasers (A) can move along the side. Perpendicular to these lasers, a camera (B) is directed towards the cube. The indenter, including four pressure sensors, can move up- and downwards.

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Figure 9: Raw data showing the force versus time signals for different indentation speeds. It was important for the accuracy of the data that the range of the pressure sensor was sufficiently large. Moreover, it was relevant to work with an indentation speed that induced initial elastic deformation.

Sunflower oil was used as the content for the container to keep a matching refractive index and density between the cube and its surroundings. This is done, in order to allow the incident beams of the two lasers to be able to find their way through the reservoir towards the gel matrix. Moreover, the cube should stay intact and not dissolve in its surroundings. In order to prevent the cube from dissolving, it was ensured the cube would only stay inside the container for a limited amount of time (only during the indentation experiment, which means less than one hour).

3.2.3

Experimental settings

During the experiments, the (50 mm diameter) indenter was moving downwards with a speed of 2.16 mm/min. In order to study a combined viscoelastic and plastic response, the speed of the indenter should be low enough to induce initial elastic deformation without exceeding the limit of the pressure sensors. The decision for this indentation speed was based on test results including speed variations (see the results in Figure 9).

The lasers move with a forward speed of 22.2 mm/s and a backward speed of 7.4 mm/s along the z-direction. The temporal and spatial resolution is optimized by focusing on the middle 25 percent of the cube (in the Z-direction). By scanning only part of the cube, information on the rest of the cube is lacking. However, the results would still show the full range of the cube in X and Y direction. The combination of an optimal time and spatial resolution together with a full X and Y range was therefore preferred over a full X, Y and Z range with sub-optimal resolution settings.

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3.3

Data retrieval and analysis methods

3.3.1

Data Retrieval: Pressure readings

Four pressure sensors were attached to the indenter. Those sensors measured the response of the cube to the compressive force. The signal was transferred directly to a PC. In a program called LabJack, the data shows the measured force signal evolving in time. The pressure signal could be used to analyze the relation between the applied compressive stress and the strain experienced by the cube.

To determine the corresponding force in Newton, a gauge was calculated by indenting a scales. For further analysis, the signal of the four sensors was averaged.

3.3.2

Data Retrieval: Real-space data

The camera recorded 1280x1024 pixel images every 0.2 mm in the Z-direction. The laser speed implied that each individual frame was captured every nine second in the forward movement of the lasers. The frames captured in the backward movement were not used in the analysis, because of the phase difference induced by the higher speed of the lasers. The frames in the forward movement was selected from the data after the experiment.

3.3.3

Pressure readings and force drop size distributions

The methods to analyze the retrieved data are described in further detail in the next para-graphs. The analysis is two-fold: on the one hand the pressure sensor data was used to obtain a force drop distribution, and on the other hand the real-space data provided a 2D displace-ment field which was converted into a strain field. We will first focus on the pressure sensor data. The analysis of the real-space data is discussed in Section 3.3.4.

As mentioned in Section 3.3.1, the signal of the four pressure sensors is averaged and binned in time. See Figure 10for an example of this signal.

Further analysis consisted of locating local force drops in the first part of the indentation. Relevant to mention is the fact that the analysis focused on the first part of the indentation, where elastic behavior dominates. Keeping in mind that plastic events do occur in this part, this regime could still be compared to the Hertzian non-linear model for elastic behavior.12

A local force drop is defined as a sudden decrease in the force signal, followed by a steep increase. After the local force drops were located, which imply local plastic events, these force drops can be plotted in a force drop distribution. At this point of the analysis the deformation behavior, with the occurrence of local plastic events in particular, could be compared to the universal power-law distribution (see Section 2 on universal deformation behavior).

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Figure 10: The pressure sensors measure the force exerted by the cube as a response to the indentation, as a function of indentation distance. The data displayed is the average signal of a cube with 0.2 VF hard inclusions.

3.3.4

Preprocessing Real-Space Data

The 2D images obtained by the camera are used to map the internal deformation of the cube. By locating certain features, specifically the dye sprinkles, and analyzing their trajectories, we are able to obtain a displacement field and a strain field representing specific moments of the indentation.

By applying a preprocessing Gaussian bandpass filter on the images, short-wavelength noise can be removed and long-wavelength variations subtracted from the image. Relevant param-eters for this filter are the size of the Gaussian kernel, the size of the rolling average and the truncation size. In the Figure below (11), an example of this preprocessing bandpass filter is shown. However, this bandpass filter would also remove the dye sprinkles. Therefore it was decided not to apply any bandpass filter on the images before the locating.

Furthermore, sections with a high mass were also caused by the shadows of the laser beams and the beads inside the cube. Therefore, mass restrictions were applied to the results after filtering, while at the same time making sure the features of the dye sprinkles remained. It should be noted that the shadows around the PMMA inclusions as well as horizontal lines due to the laser beams remain a challenge in the feature locating.

3.3.5

Particle locating and tracking

The analysis described below is performed on a set of filtered 2D frames of the cube (in the middle of the cube): specifically the frames at z-location Z, Z-3 and Z+3. By summing the features found in the multiple 2D frames, we correct for missing features due to a phase shift of the laser positions (which could happen if only analyzing a single 2D frame).

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(a) Original picture. (b) Filtered picture.

Figure 11: The preprocessing bandpass filter removes short-wavelength noise and subtracts long-wavelength variations. This filter is applied on the 2D real-space images of the cubes before analyzing the deformation.

To locate the features the open-source particle-tracking toolkit Trackpy was used, based on the algoritm by Crocker and Grier.52 Relevant parameters in this part of the analysis are the

diameter of the feature, the minimal mass (in terms of light intensity - contrast to the back-ground), and the noise size. With the linking function of the Trackpy package, the various features are linked to features subsequent in time. The search range (in pixels), memory of disappeared particles (in frames) and minimal trajectory length (in frames) are relevant pa-rameters for those steps. The Python trajectory function eventually provides the trajectories of the located features. Moreover, the linked features are used to create a displacement and strain field inside the cube, see the next paragraph.

Schuster et al. (2016)53 provide a description on the static and dynamic errors in particle

tracking and make suggestions to avoid these errors. The subject of interest was not related to our real-space data. However we were still able to implement their suggestions, since the analyzing method was very similar. The first step is to follow the features manually over a range of multiple time frames. This gives a starting point for the parameters such as the maximum displacement of one single feature. Also, it can give an impression of the quality of the feature locating.

As a second check-up for locating accuracy, the eccentricity and the mass versus size of the features can be plotted (see Figure 12).

Plots like these can reveal features which have an anomalous ratio between size and ec-centricity or between size and mass. This data can be used to decide to remove all features

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Figure 12: Mass versus size of the annotated features. Plotting these relations can help omitting certain falsely annotated features.

with a ratio above or below a certain threshold.

3.3.6

Local strain field

To calculate the nonaffine strain inside the cube, the method was used as described by Falk and Langer17(see also Rahmani et al., 201312). This is done by comparing the displacement of

neighbouring particles to the displacement if it would have been a uniform (affine) deformation. The local strain tensor is the symmetric part of the deformation tensor,12 here:

ij =

1

2(Hij+ H

T

ij) (16)

Here, H is the deformation tensor. The deformation tensor was computed from the defor-mation of the features annotated by the Trackpy function. By defining a neighbouring region as the cut-off value, and comparing the trajectory of a feature to its neighbours, the nonaffine deformation could be computed. Both the local maximum strain as well as the compressive strain could be analyzed with the calculated local strain tensor. As mentioned by Rahmani et al. in 2013,3 the maximum strain is an invariant of the strain tensor and is an appropriate

measure of "the inhomogeneous deformation of isotropic materials".3 This maximum strain is

calculated by taking the largest and smallest eigenvalues of the strain tensor (1 and 2):

max =

1

2 | 1− 2 | (17)

The compressive strain is found by taking the yy components of the strain tensor (which

is the indentation direction). For the final strain fields, the edges of the cubes were not taken into account. At the edges, the features did not have sufficient neighbours and therefore no physically relevant strain values could be computed. As mentioned above, the analysis of the deformation inside the cube focused on the 2D-plane parallel to the indentation direction (the XY-plane in the used coordinate system). In addition, the plane perpendicular to the

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indentation direction (i.e. the XZ-plane) was analyzed. At a certain height inside the cube (i.e. a certain Y-value), the computed strain data of the single XY-planes was averaged on the X-axis. Taking all these results of subsequent Z-frames together, a strain field in the XZ-plane was created (see Figure 13).

Figure 13: By taking strain data from the subsequent XY-planes, a XZ-plane at a certain Y-value is obtained. This plane, perpendicular to the indentation direction, can be used to analyze the deformation throughout the entire frame.

3.4

Discussion of methods

There are a couple of factors which could influence the quality of the data. Firstly, the layered structure of the cubes could lead to weaker regions between the layers, which could affect the overall deformation of the cube. The quality of the pressure readings was improved with practical techniques. For example, the wires from the sensors to the computer were shielded to limit the noise signal.

Moreover, the beads inside the cubes could cause reflection of the laser lights, making it harder to distinguish between actual features and noise during the locating. Although attempts have been made to optimize the locating and tracking quality, it should be noted that the strain fields might include wrongly annotated features. In a research conducted by Chenouard et al.54 different particle tracking methods are revised. Both the spatial (i.e. locating a

par-ticle in a particular frame) and the temporal (i.e. creating trajectories by coupling features in multiple frames) aspect are taken into account. One of the most important properties of well-functioning tracking methods is the explicit usage of knowledge about the particle motion. Working beyond nearest neighbour model also increased the success rate of a method. In the paper by Chenouard et al.54 a retrospective analysis of the distribution of localization errors

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is presented.

The uncertainty within particle tracking can be estimated following a method by Savin and Doyle.55 This could be used in future research.

Notwithstanding its limitations, the obtained strain fields do correspond to the observed non-affine displacements inside the cube and therefore have a significant physical meaning. In future research it could be useful to apply 3D locating to obtain even more insights in the deformation mechanism. In order to apply 3D locating it would be necessary to increase the Z-resolution of the data.

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4

Results

In this Section the results of the indentation experiments are presented. Viscoelastic cubes with PMMA inclusions covering a volume fraction between 0 and 40% were indented. In Subsection 4.1, we look at the force signal and the force drop size distributions. The real-space data is analyzed in the next Subsections: in 4.2.1 the local maximum and compressive strain field of the cube without inclusions is presented, followed by a cube with φinc = 0.1 in 4.3 and a cube with φinc = 0.2 in

4.4. Moreover, we focus on the clustering of local strain regions in Subsection 4.5 and compare the strain distributions to Hertzian elastic theory in Subsection 4.6..

4.1

Force drop size distributions

We investigate the deformation under indentation by measuring the force exerted on the indenter. As discussed in Section 3.3.3 the signal of four pressure signals is retrieved during the indentation experiment. The stress versus strain curves show the build-up of internal stress as the material is being indented. In Figure 14 the binned pressure readings of the 0.0, 0.1, 0.2 and 0.4 VF cubes are displayed (showing the force in Newton versus indentation distance). We define a force drop as a sudden decrease in the force signal, followed by a steep increase. From Figure 14it can be seen that in the cube without beads (Figure14a) the force increases more gradually than in the cubes with beads. Moreover, more large pressure drops seem to occur in the 0.1 VF and 0.2 VF cubes.

In Figure15, this force drop size distributions are plotted on a log-log scale (for the analysis method the reader is referred to Section 3). The distributions show similar behavior for the different cubes as they follow the same power-law with an exponent of -3/2.

The overlapping force drop distributions are surprising, since the materials contain different volume fractions of hard inclusions. One could have expected that the hard inclusions stop the avalanches. In that scenario, a strong dependence of the drop size distributions on the inclusion volume fractions should have become visible. The data shown in Figure 15 could possibly imply that the avalanches are not affected by the presence (and the amount of) hard inclusions.

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(a) 0.0 VF (b) 0.1 VF

(c) 0.2 VF (d) 0.4 VF

Figure 14: Binned pressure readings for the cubes with 0.0, 0.1, 0.2 and 0.4 VF beads.

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4.2

Real-space deformation of the beadless cube

4.2.1

2D strain distributions

While the power-law distributions are similar to vastly different systems, the intriguing question is how the local deformation inside our specific composite facilitates the observed stress-relaxation events. We therefore simultaneously image the interior of the cube and observe a clear structure as a result of local high dye concentrations. These dye concentrations enable us to analyze internal deformations. Figure 16 offers a 2D view at several moments during the indentation (in the XY-plane). The spherical form of the indenter becomes visible in the upper part of the cube. Moreover, horizontal lines can be observed which are caused by the layered fabrication process of the cube (see Section 3.2.1).

10 mm (a) T=0sec. 10 mm (b) T=45sec. 10 mm (c) T=135sec. 10 mm (d) T=180sec.

Figure 16: 2D images of the cube (φinc=0.0) recorded at several moments after indentation start.

The red box in 16dindicates the area used to calculate the 2D strain fields.

In Figure17the trajectories and the local compressive as well as the local maximum strain fields are depicted in 2D. For the analysis, certain filters had been applied to limit the impact of the horizontal bright lines. This resulted in the empty spaces in the strain fields. Throughout

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the indentation the maximum strain is the highest in the upper part of the cube. Moreover, a pattern of compression (indicated in blue) and dilation (in red) can be observed in the compressive strain plots. A possible explanation for the dilation can be tracking error. This pattern fluctuates heavily, but strain correlations are visible on short scale. These correlations indicate the existence of locally weak spots.

x [px] y [px] (a) T=36 to 45 sec. 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (b) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (c) max x [px] y [px] (d) T=81 to 90 sec. 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (e) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (f ) max x [px] y [px] (g) T=126 to 135 sec. 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (h) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (i) max

Figure 17: Trajectory of the located features inside the cube (φinc=0.0) (17a,17dand 17g). Both

the compressive (17b, 17e and 17h) as well as the local maximum strain (17c, 17f and 17i) are calculated incrementally (in time intervals [36-45], [81-90] and [126-135] sec. after the indentation start). The 2D binned results are shown in a XY-frame.

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4.2.2

Taking a perpendicular view

The strain fields inside the cube with no inclusions show local correlations. However, they lack the longer scale gradient which is known from a purely elastic response. To further investigate this observed deviation from elastic behavior, we use the unique 3D imaging capacity of our experimental setup to reconstruct a displacement field in the XZ-plane (perpendicular to the indentation direction): see Figure 18. The strain values were calculated per z-frame (again computed from z-3, z and z+3). The level of the XY-bands was chosen explicitly to include the depth of the predicted maximum strain: ranging from 7.5 to 17.5 mm distance the top side of the cube. For interpretation purposes, two different bin sizes are included per frame. In Figures 18a and 18c it can be seen that the compressive strain dominates in the middle and upper part (blue areas). This corresponds to the centre of the cube, where the impact of the indenter is the largest. This area further develops in the later stage (at 135 seconds, see Figures 18c and 18d). Further away from the center, regions of local compression and dilation become apparent. A remarkably dominant region of compression is observed in the bottom left corner of Figure 18a. Interestingly, this area is not located directly underneath the indenter. Later in time, this area seems to clear away (Figure 18c).

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x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (a) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (b) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (c) T=135sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (d) T=135sec.

Figure 18: Perpendicular view of the compressive strain [in 0.0 VF cube] at T=45 sec. (18a and

18b) and at T=135 sec. (18cand18d). Strain values were taken from the range of 7.5 to 17.5 mm below the original upper surface of the cube. In these images, a Z-X view is provided.

4.3

Filling the cube with 0.1 VF inclusions

4.3.1

2D strain distributions

In this section we analyze a cube with φinc = 0.1. As the indenter proceeds, internal

defor-mations become visible. This can be seen in Figure 19. Figure 19d shows that the bottom part of the cube becomes affected by the indentation and the material is pushed outwards.

The local dye spots enable us to track the motion of these regions and reconstruct the trajectories, which are depicted in Figure 20. Areas of high strain seem to develop around the location of the PMMA inclusions, as can be seen in the plots showing the local maximum and compressive strain field. Yellow arrows in the compressive strain fields indicate the location of some of the inclusions.

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10 mm (a) T=0sec. 10 mm (b) T=45sec. 10 mm (c) T=135sec. 10 mm (d) T=180sec.

Figure 19: 2D images of the cube (φinc= 0.1) recorded at several stages of the indentation process.

The red box in 19cindicates the area used for the perpendicular strain views.

4.3.2

Fluctuating strain field in 3D

To gain further insight in the deformation of the cube, we again visualize the strain perpendic-ular to the indentation direction (see Figure21). Strain values were taken from an indentation depth of 11.25 to 21.25 mm below the surface of the cube. For interpretation purposes, two different bin sizes are included per frame.

In Figure 21 we observe a high strain field centered around the indenter tip, however fluctuating in time. Specifically, remarkably high amount of dilation (red areas) at T=45 sec. can be seen. However, this changes drastically toward T=135 sec. Here we observe more compression (blue areas, visible in Figures 21c and 21d).

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x [px] y [px] (a) T=36 to 45 sec. (b) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (c) max x [px] y [px] (d) T=81 to 90 sec. (e) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (f ) max x [px] y [px] (g) T=126 to 135 sec. (h) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (i) max

Figure 20: Trajectory of the located features inside the cube (φinc=0.1) (20a,20dand 20g). Both

the compressive (20b, 20e and 20g) as well as the local maximum strain (20c, 20f and 20i) are calculated incrementally. The 2D binned results are shown in a XY-frame in the middle of the cube. Yellow arrows in the compressive strain fields indicate the location of some of the inclusions.

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x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (a) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (b) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (c) T=135sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (d) T=135sec.

Figure 21: Perpendicular view of the compressive strain [in 0.1 VF cube] at T=45sec. (21a and

21b) and at T=135sec. (21cand 21d). Strain values were taken from the range of 11.25 to 21.25 mm below the original upper surface of the cube

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4.4

Increasing the PMMA inclusions to 20%

4.4.1

2D strain distributions

From the first results of a cube containing 10% hard inclusions, we observed that more areas of local dilation became apparent. Now, we are interested to analyze what the effect is of adding more PMMA inclusions. The 2D images of a cube with φinc = 0.2 at several moments

during the indentation are depicted in Figure 22. In Figure 23 the development of the strain fields is depicted. Yellow arrows indicate the location of some of the inclusions present inside the cube. We observe the development of local strain maxima: in Figure23ha prominent area of compression become visible just underneath the indenter tip (indicated by a green circle).

10 mm (a) T=0sec. 10 mm (b) T=45sec 10 mm (c) T=90sec. 10 mm (d) T=135sec.

Figure 22: 2D frames (XY plane) of the cube (φinc=0.2), at T= 0, 45, 90 and 135 seconds after

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x [px] y [px] (a) T=36 to 45 sec. (b) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (c) max x [px] y [px] (d) T=81 to 90 sec. (e) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (f ) max x [px] y [px] (g) T=126 to 135 sec. (h) yy 0.000 0.005 0.010 0.015 0.020 0.025 0.030 (i) max

Figure 23: Trajectory of the located features inside the cube (φinc=0.2) (23a,23dand 23g). Both

the compressive (23b, 23e and 23h) as well as the local maximum strain (23i, 23f and 23i) are calculated incrementally (in the interval between 36 and 45 seconds). The 2D binned results are shown in a XY-frame. The yellow arrows indicate some of the inclusions present inside the cube. The green circle in Figure 23hdenotes a high compression peak underneath the indenter tip.

4.4.2

More strain fluctuations in the perpendicular plane

The compressive strain field perpendicular to the indentation direction is depicted in Figure

24. Strain values were taken from an indentation depth of 3.75 to 13.75 mm below the surface of the cube. For interpretation purposes, two different bin sizes are included per frame.

In Figure 24it can be seen that at T=45 sec., areas of compression dominate. Later, at T=135 sec., more areas of dilation become apparent. This could be a sign that the initial compressive strain (due to the impact of the indenter) later gives room to local dilation, probably triggered by the beads present inside the cube. Also, we look at an area deeper in the cube: from 9.4 to 18.8 mm depth. It can be seen in Figure 25 that this area covers the locations of the expected Hertzian maximum strain at the considered moments in time. We will refer back to these Hertzian predicted strain values in Subsection 4.6. We observe large areas of compression in Figure 26. In particular in the images on the right side (smaller bin size) the data shows that areas of dilation develop between T=45 sec. (Figure 26b) and

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x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (a) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (b) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (c) T=135sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (d) T=135sec.

Figure 24: Perpendicular view of the compressive strain (φinc=0.2) at T=45 sec. (24a and 24b)

and at T=135 sec. (24c and24d). Strain values were taken from the range of 3.75 to 13.75 mm below the original upper surface of the cube.

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10 mm

Figure 25: 2D Image of the cube (φinc), where the red box indicates the area from 9.4 to 18.8

mm below the surface. This area is used for the perpendicular strain view in Figure 26. The yellow lines indicate the locations of the predicted maximum strain at T=45, 90 and 135 sec. (from top to bottom) after the indication start.

x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (a) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (b) T=45sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (c) T=135sec. x-axis z-axis 0.03 0.02 0.01 0.00 0.01 0.02 0.03 (d) T=135sec.

Figure 26: Perpendicular view of the compressive strain (φinc = 0.2) at T=45 sec. (24a and24b)

and at T=135 sec. (24cand24d). Strain values were taken from the range of 9.4 to 18.8 mm below the original upper surface of the cube.

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4.5

Clustering of local strain regions

Throughout the research, the existence and clustering of local strain regions has been a central point of attention. No clear constant strain field appeared during the indentation of the composites. However, we did observe the distribution of local strain events throughout the material. Figure 27 shows the trajectories of the cube with φinc = 0.1 in subsequent time

frames. Also, the normalized maximum strain is plotted in 2D. Here, areas with similar strain values are marked with the same color. It can be seen that clusters of similar strain are formed in subfigure d, and grow larger in subfigure e and f. This indicates that areas of local weak spots exist, in which the strain is locally similar.

Figure 27: Clustering of local strain regions in the cube (φinc=0.1). Depicted are strain fields

subsequent in time. It can be observed that clusters of similar strain are formed in subfigure d, and grow larger in subfigure e and f.

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